Three nodal solutions of singularly perturbed elliptic equations on domains without topology

We prove the existence of three nodal solutions of the Dirichlet problem for the singularly perturbed equation -εΔ u + u = f (u) for ε > 0 small on any bounded domain Ω ⊂ R-N. The nonlinearity f grows superlinearly and subcritically. We do not require symmetry conditions nor conditions on the geometry or the topology of the domain. Two solutions have precisely two nodal domains, and the third solution has at most three nodal domains. A corresponding result holds true for the semilinear equation -Δ u + u = f (u) on Ω provided Ω contains a large ball. © 2005 Elsevier SAS. All rights reserved.